Empirical Statistical Mechanics · 2008-09-25 · Empirical Statistical Mechanics Wang, Xiaoming...

Empirical Statistical Mechanics

Wang Xiaomingwxmmathfsuedu

Florida State University

Stochastic and Probabilistic Methods in Ocean-AtmosphereDynamics Victoria July 2008

Wang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics

Example 1 Large Hamiltonian systemLarge (N 1) particles with position xj = qj and momentum pj(j = 1 middot middot middot N)

Least action principle

A =

int t

t0L(x(s) x(s) s) ds

Lagrange equations of motion

partLpartxj

minus ddtpartLpartxj

= 0

HamiltonianH =

sumj

pj xj minus L pj =partLpartxj

Hamiltonian system (common notation qj = xj )

dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj

Conservation of the Hamiltonian HWang Xiaoming wxmmathfsuedu Empirical Statistical Mechanics



A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj


Example 2 Logistic map

T (x) = 4x(1minus x) x isin [01]

Figure Sensitive dependence

0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

07

08

09

1

Orbits of the Logistic map with close ICs

Figure Statistical coherence

-02 0 02 04 06 08 1 120

100

200

300

400

500

600


Example 3 Lorenz 96 model

Edward Norton Lorenz 1917-2008

Lorenz96 duj

dt= (uj+1 minus ujminus2)ujminus1 minus uj + F

j = 01 middot middot middot J J = 5F = minus12

Figure Sensitive dependence Figure Statistical coherence

minus20 minus10 0 10 200

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Lorenz96 duj




(Sensitive dependence)



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L96_demoavi
Media File (videoavi)

Example 4 Rayleigh-Beacutenard convection

1Pr

(partupartt

+ (u middot nabla)u) +nablap = ∆u + Ra kθ nabla middot u = 0 u|z=01 = 0

partθ

partt+ u middot nablaθ minus u3 = ∆θ θ|z=01 = 0 (θ = T minus (1minus z))

Pr =ν

κ

Ra =gα(Tbottom minus Ttop)h3

νκ


Figure Lord Rayleigh (JohnWilliam Strutt) 1842-1919

Figure Henri Beacutenard (left)1874-1939


RBC set-up and numerics

Figure RBC set-up Figure Numerical simulation

(infin Pr simulation)


m1mpg
Media File (videompeg)

Statistical Approaches

dudt

= F(u) u isin H

S(t) solutionsemigroup

Long time average

lt Φ gt= limTrarrinfin

1T

int T

0Φ(v(t)) dt

Statistical coherence in terms of histogram implies independenceon initial dataSpatial averages

lt Φ gtt=

intH

Φ(v) dmicrot(v)

microt t ge 0 statistical solutionsFinite ensemble average vj(t) = S(t)v0j j = 1 middot middot middot N

lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))


Statistics Solutions

microt t ge 0 dvdt

= F(v) S(t) t ge 0

Pull-backmicro0(Sminus1(t)(E)) = microt(E)

Push-forward (Φ suitable test functional)intH

Φ(v) dmicrot(v) =

intH

Φ(S(t)v) dmicro0(v)

Finite ensemble example

micro0 =Nsum

j=1

pjδv0j (v) microt =Nsum

j=1

pjδvj (t)(v)vj(t) = S(t)v0j



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1



Relative stability of statistical properties

Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation

|microt(E)minus microt(E)| le |micro0(Sminus1(t)E)minus micro0(Sminus1(t)E)|



Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation



Liouvillersquos equations

Figure Joseph Liouville1809-1882

Liouville type equation

ddt

intH

Φ(v) dmicrot(v)

=

intHlt Φprime(v)F(v) gt dmicrot(v)

Φ good test functionals eg

Φ(v) = φ((vv1) middot middot middot (vvN))

Liouville equation (finite d)

part

parttp(v t) +nabla middot (p(v t)F(v)) = 0





ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part



Hopfrsquos equations

Figure Eberhard Hopf1902-1983

Hopfrsquos equation (special case of Li-ouville type)

ddt

intH

ei(vg) dmicrot(v)

=

intH

i lt F(v)g gt ei(vg) dmicrot(v)





ddt

intH

ei(vg) dmicrot(v)

=

intH



Invariant measures (IM)(Stationary StatisticsSolutions)

Invariant measure (IM) micro isin PM(H)

micro(Sminus1(t)(E) = micro(E)

Stationary statistical solutions essentiallyintHlt F (v)Φprime(v) gt dmicro(v) = 0forallΦ












Birkhoffrsquos Ergodic Theorem

Figure George DavidBirkhoff 1884-1944 Definition

micro is ergodic if micro(E) = 0 or 1 for allinvariant sets E

Theorem (Birkhoffrsquos ErgodicTheorem)

If micro is invariant and ergodic thetemporal and spatial averages areequivalent ie

limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =

intHϕ(u) dmicro(u) as


Examples of IM 1 dissipative

drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1

Non-uniqueness of invariant measureSupport of invariant measure may be singularQuestion of physical relevance



drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1



Examples of IM 2 Hamiltonian

Hamiltonian system with energy H(pq) (q = x)Canonical measuredistribution

dmicro(pq) = Zminus1 exp(minusβH(pq)) dpdqZ =

intH

exp(minusβH(pq)) dpdq

Z partition function β inverse temperatureMicro-canonical measuredistribution

dmicro = χminus1 exp(minusβH(pq))δ(E minus H(pq)) dpdq

χ =

intexp(minusβH(pq))δ(E minus H(pq)) dpdq = exp(

Sk

)

χ structure function k Boltzmann const





intH




χ =


Sk

)






intH




χ =


Sk

)



Shannonrsquos entropy (measuring uncertainty)

Figure Claude Shannon1916-2001

Definition (Shannon entropy)

S(p) = S(p1 pn) = minusnsum

i=1

pi ln pi

p isin PMn(A) probability measureon A = a1 an






i=1

pi ln pi



Uniqueness of Shannon entropy

Figure Edwin ThompsonJaynes 1922-1998

Theorem (Jaynes)

Hn(p1 pn) continuousA(n) = Hn(1n 1n)monotonic in nA = a1 an = A1

⋃A2

A1 = a1 akA2 = ak+1 anw1 = p1 + middot middot middot+ pk w2 = pk+1 + middot middot middot+ pn

Hn(p1 pn) = H2(w1w2)

+w1Hk (p1w1 pkw1)

+w2Hnminusk (pk+1w2 pnw2)

Then existK gt 0 st Hn(p1 pn) =KS(p1 pn) = minusK

sumnj=1 pj ln pj


Jaynesrsquo maximum entropy principleGiven statistical measurements Fj of given functions fj j = 1 r le n minus 1

Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r

or continuous version

Fj =lt fj gtp=

intfj(x)p(x) dx

Definition (Empirical maximum entropy principle)

Least biased most probable pdf plowast is given by

maxpisinCS(p) = S(plowast) plowast isin C

C = p isin PM(A)∣∣ lt fj gtp= Fj 1 le j le r

plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)

Z partition function



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)



Maximum relative entropy principle

Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

with prior distribution p0

Relative entropy as semi-distance

minusS(pp0) ge 0forallp

Definition (Empirical maximum relative entropy principle)


maxpisinCS(pp0) = S(plowastp0) plowast isin C

for a given set of constraints C defined by




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)










Example discrete case

No constraint The one that maximizes the Shannon entropy onA = a1 middot middot middot aN is the uniform distribution on ANo constraint but with prior p0 The least biased (most probable)one is the prior





Maximum entropy principle for continuous pdf

Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)

Π0(x)dx = minusP(pΠ0)

with prior distribution Π0



maxpisinCS(pΠ0) = S(plowastΠ0) plowast isin C





Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)









Example continuous case

The most probable state with given first and second moments onthe line is the GaussianHamiltonian system with energy H being the only conservedquantity the most probably state is the Gibbs measure(macro-canonical measure)

1Z

exp(minusβH)




1Z

exp(minusβH)


Maximum entropy for continuous fieldOne point statistics for q(x) x isin Ω

(application to PDE)

ρ(x λ) ge 0intρ(x λ) dλ = 1ae

int q+

qminusρ(x λ) dλ = Probqminus le q(x) lt q+

lt F (q) gtρ=1|Ω|

intΩ

intR1

F (x λ)ρ(x λ) dλdx

S(ρ) = minus 1|Ω|

intΩ

intR1ρ(x λ) ln ρ(x λ) dλdx

S(ρΠ0) = minus 1|Ω|

intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx

Maximum entropy principle remain the same





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx



Barotropic quasi-geostrophic equation with topography

Barotropic quasi-geostrophic equation

partqpartt

+nablaperpψ middot nablaq = 0 q = ∆ψ + h

q potential vorticity ψ stream-function h bottom topographyΩ = (02π)times (02π) per bcConserved quantities kinetic energy E and total enstrophy E

E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx

There exist infinitely many conserved quantities




partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx



Application of maximum entropy principle to barotropicQG

Energy and total enstrophy as constraints

E(ρ) = minus 12|Ω|

intΩ

ψ(q minus h) q(x) =

intR1λρ(x λ)dλ q = ∆ψ + h

E(ρ) =12

int intR1λ2ρ(x λ) dλdx

Lagrangian multiplier calculation

δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ

δintρ(x λ) dλδρ

= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx


Application of maximum entropy principle to barotropicQG continued

Most probable state ρlowast

minus1minus ln(ρlowast) = minusθλψlowast +α

2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)

θ α Lagrange multipliers for energy and enstrophy γ(x)Lagrange multipliers for the pdf constraintMean field equation

∆ψlowast + h =θ

αψlowast = microψlowastwith micro =

θ

α α gt 0

Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable(Arnold-Kruskal stability)





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0



Application of maximum relative entropy principle tobarotropic QG

Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)

θ Lagrange multiplier for energyMean field equation

∆ψlowast + h =θ


θ

α α gt 0

Under generic topography existmicro = micro(E) isin (minus1infin) ψmicro solves themean field equation uniquely and is nonlinearly stable



Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0



Energy circulation theory with a general prior

BQG with bottom topography and channel geometryΩ = (02π)times (0 π)

ψ = ψprime =sumkge1

sumjge0

(ajk cos(jx) + bjk sin(jx)) sin(ky)

Energy and circulation as conserved quantities


intΩ

ψωdx Γ(ρ) =1|Ω|

intΩ

intλρ(x λ)dλdx

Most probable one-point statistics

ρlowast(x λ) =e(θψlowastminusγ)λΠ0(x λ)int

e(θψlowastminusγ)λΠ0(x λ) dλ

θ γ Lagrange multipliers for energy and circulation respectivelyMean field equation

∆ψlowast+h =1θ

(part

partψlnZ(θψ x)

) ∣∣∣∣ψ=ψlowast

Z(ψx) =

inte(ψminusγ)λΠ0(x λ) dλ





sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =



Mean field statistical theory for point vortices

2D Euler in a disc Ω = BR(0)Energy circulation and angular momentum (A = 1

|Ω|intΩ|x|2ω) as

conserved quantitiesAppropriate prior distribution

Nsumi=1

1N

(δω0(λ)otimes δxi ) δω0(λ)dx|Ω|

= Π0(λ)

Mean field equation for the most probable state

∆ψlowast = ωlowast =|Ω|

intλe(θψlowastminusγminusα|x|2)λΠ0(x λ) dλint int

e(θψlowastminusγminusα|x|2)λΠ0(x λ) dλ dx

Case with no angular momentum

∆ψlowast =Γ0eminusβψ

lowastinteminusβψlowastdx

β = minusθω0

Case with angular momentum


lowastminusα|x|2intBR(0)

eminusβψlowastminusα|x|2dx




|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0






Summary empirical statistical mechanics

Undampedunforced setting customaryInformation theoretical approach maximize Shannon entropy orrelative entropy with prior and given measurements (conservedquantities)Conserved quantities become constraints on the one-pointstatistics ρMean field equation

q = G(ψ)

Most of them (large scale structure) are stable under appropriateassumptions and hence physically relevantThe process is relatively easy and versatile (with given prior andconserved quantities)Further reading Majda and W Nonlinear Dynamics andStatistical Theories for Basic Geophysical Flows CUP 2006




q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj





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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj





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Lorenz96 duj





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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj





0 20 40 60 80 100 120 140 160 180 2000

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02

03

04

05

06

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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj





0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

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08

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1



-02 0 02 04 06 08 1 120

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Lorenz96 duj





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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj




A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj





0 20 40 60 80 100 120 140 160 180 2000

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02

03

04

05

06

07

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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





A =

int t

t0L(x(s) x(s) s) ds


partLpartxj


= 0

HamiltonianH =

sumj



dxj

dt=partHpartpj

dpj

dt= minuspartH

partxj





0 20 40 60 80 100 120 140 160 180 2000

01

02

03

04

05

06

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08

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1



-02 0 02 04 06 08 1 120

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Lorenz96 duj





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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






0 20 40 60 80 100 120 140 160 180 2000

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02

03

04

05

06

07

08

09

1



-02 0 02 04 06 08 1 120

100

200

300

400

500

600




Lorenz96 duj





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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





Lorenz96 duj





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Lorenz96 duj







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L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





Lorenz96 duj







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4000

6000

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10000

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14000


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6000

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10000

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14000


L96_demoavi


1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




1Pr

(partupartt


partθ


Pr =ν

κ


νκ









m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)










m1mpg


dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




dudt

= F(u) u isin H


Long time average


1T

int T

0Φ(v(t)) dt


lt Φ gtt=

intH

Φ(v) dmicrot(v)


lt Φ gtt=1N

Nsumj=1

Φ(vj(t))



microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




microt t ge 0 dvdt

= F(v) S(t) t ge 0



Φ(v) dmicrot(v) =

intH



micro0 =Nsum

j=1


j=1




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Finite ensemble1N

Nsumj=1

Φ(uj(t))

Total variation






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






ddt

intH

Φ(v) dmicrot(v)

=





part






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






ddt

intH

ei(vg) dmicrot(v)

=

intH






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






ddt

intH

ei(vg) dmicrot(v)

=

intH























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)























limTrarrinfin

1T

int T

0ϕ(S(t)u) dt =




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




drdt

= r(1minus r2)dθdt

= 1

micro1 = δ0

dmicro2 =1

2πdθ on r = 1






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






intH




χ =


Sk

)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






intH




χ =


Sk

)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






intH




χ =


Sk

)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)







i=1

pi ln pi







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)







i=1

pi ln pi





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





Theorem (Jaynes)


⋃A2



+w1Hk (p1w1 pkw1)



sumnj=1 pj ln pj



Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Fj =lt fj gtp=nsum

i=1

fj(ai)pi j = 1 r


Fj =lt fj gtp=

intfj(x)p(x) dx





plowasti =exp

(minus

sumrj=1 λj fj(ai)

)Z

Z =nsum

i=1

exp

minus rsumj=1

λj fj(ai)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)











Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Relative entropy

S(pp0) = minusNsum

j=1

pj lnpj

p0j

= minusP(pp0)

















Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)










Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Entropy

S(p) = minusint

p(x) ln p(x) dx

Relative entropy

S(pΠ0) = minusint

p(x) lnp(x)











1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





1Z

exp(minusβH)




1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





1Z

exp(minusβH)





int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






int q+


lt F (q) gtρ=1|Ω|

intΩ

intR1


S(ρ) = minus 1|Ω|

intΩ



intΩ

intR1ρ(x λ) ln

ρ(x λ)

Π0(x λ)dλdx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





partqpartt



E = minus 12|Ω|

intΩ

ψ∆ψ dx

E =1

2|Ω|

intΩ

q2 dx






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






intΩ



E(ρ) =12



δSδρ

= minus1minus ln ρ

δEδρ

=12λ2

δEδρ

= λψ(x)δqδρ

= λ


= δx





2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






2λ2 + γ(x)

ρlowast(x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)




Energy constraint

E(ρ) = minus12

intΩ



Gaussian prior

Π0(λ) =

radicαradic2π

exp(minusα2λ2)

Most probable state

ρlowast(~x λ) =

radicαradic2π

exp(minusα2

(λminus θ

αψlowast)2)


∆ψlowast + h =θ


θ

α α gt 0






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)






sumjge0




intΩ


intΩ

intλρ(x λ)dλdx





∆ψlowast+h =1θ

(part

partψlnZ(θψ x)


Z(ψx) =





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





|Ω|intΩ|x|2ω) as


Nsumi=1

1N


= Π0(λ)








β = minusθω0








q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)





q = G(ψ)



Empirical Statistical Mechanics · 2008-09-25 · Empirical Statistical Mechanics Wang, Xiaoming...

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Transcript of Empirical Statistical Mechanics · 2008-09-25 · Empirical Statistical Mechanics Wang, Xiaoming...